Classical transients and the support of open quantum maps

TL;DR

This paper investigates the classical structures supporting quantum open systems, revealing that the classical repeller may not always be the relevant invariant object, especially in short-time dynamics, challenging existing semiclassical theories.

Contribution

The study provides numerical evidence that the classical repeller is not always the support for quantum resonances in open systems, highlighting the need for revised semiclassical descriptions.

Findings

Eigenvalue distribution follows fractal Weyl law

Short periodic orbit theory fails for some maps

Classical repeller may not support quantum resonances

Abstract

The basic ingredients in a semiclassical theory are the classical invariant objects serving as a support for the quantization. Recent studies, mainly obtained on quantum maps, have led to the commonly accepted belief that it is the classical repeller -- the set of non escaping orbits in the future and past evolution -- the object that suitably plays this role in open scattering systems. In this paper we present numerical evidence warning that this may not always be the case. For this purpose we study recently introduced families of tribaker maps [L. Ermann, G.G. Carlo, J.M. Pedrosa, and M. Saraceno, Phys. Rev. E {\bf 85}, 066204 (2012)], which share the same asymptotic properties but differ in their short time behavior. We have found that although the eigenvalue distribution of the evolution operator of these maps follows the fractal Weyl law prediction, the theory of short periodic orbits for open maps fails to describe the resonance eigenfunctions of some of them. This is a strong indication that new elements must be included in the semiclassical description of open quantum systems.